TY - JOUR

T1 - Deformations of Symmetric Simple Modular Lie (Super)Algebras

AU - Bouarroudj, Sofiane

AU - Grozman, Pavel

AU - Leites, Dimitry

N1 - Funding Information:
We are thankful to A. Krutov and A. Lebedev for huge help. We are thankful to N. Chebochko and M. Kuznetsov for helpful discussions of their unpublished results pertaining to this paper. We thank A. Dzhumadildaev for pointing out [67] thus correcting an error. We are very thankful to the referees, carefully selected by SIGMA, for constructive criticism and extremely careful job. D.L. is thankful to MPIMiS, Leipzig, where he was Sophus-Lie-Professor (2004-07, when certain results of this paper were obtained), for financial support and most creative environment. We are thankful to M. Al Barwani, Director of the High Performance Computing resources at New York University Abu Dhabi for the possibility to perform the difficult computations of this research. S.B. and D.L. were supported by the grant AD 065 NYUAD.
Publisher Copyright:
© 2023, Institute of Mathematics. All rights reserved.

PY - 2023

Y1 - 2023

N2 - We say that a Lie (super)algebra is “symmetric” if with every root (with respect to the maximal torus) it has the opposite root of the same multiplicity. Over algebraically closed fields of positive characteristics (up to 7 or 11, enough to formulate a general conjec-ture), we computed the cohomology corresponding to the infinitesimal deformations of all known simple finite-dimensional symmetric Lie (super)algebras of rank < 9, except for super-izations of the Lie algebras with ADE root systems, and queerified Lie algebras, considered only partly. The moduli of deformations of any Lie superalgebra constitute a supervariety. Any infinitesimal deformation given by any odd cocycle is integrable. All deformations corresponding to odd cocycles are new. Among new results are classifications of the cocycles describing deforms (results of deformations) of the 29-dimensional Brown algebra in characteristic 3, of Weisfeiler–Kac algebras and orthogonal Lie algebras without Cartan matrix in characteristic 2. Open problems: describe non-isomorphic deforms and equivalence classes of cohomology theories. Appendix: For several modular analogs of complex simple Lie alge-bras, and simple Lie algebras indigenous to characteristics 3 and 2, we describe the space of cohomology with trivial coefficients. We show that the natural multiplication in this space is very complicated.

AB - We say that a Lie (super)algebra is “symmetric” if with every root (with respect to the maximal torus) it has the opposite root of the same multiplicity. Over algebraically closed fields of positive characteristics (up to 7 or 11, enough to formulate a general conjec-ture), we computed the cohomology corresponding to the infinitesimal deformations of all known simple finite-dimensional symmetric Lie (super)algebras of rank < 9, except for super-izations of the Lie algebras with ADE root systems, and queerified Lie algebras, considered only partly. The moduli of deformations of any Lie superalgebra constitute a supervariety. Any infinitesimal deformation given by any odd cocycle is integrable. All deformations corresponding to odd cocycles are new. Among new results are classifications of the cocycles describing deforms (results of deformations) of the 29-dimensional Brown algebra in characteristic 3, of Weisfeiler–Kac algebras and orthogonal Lie algebras without Cartan matrix in characteristic 2. Open problems: describe non-isomorphic deforms and equivalence classes of cohomology theories. Appendix: For several modular analogs of complex simple Lie alge-bras, and simple Lie algebras indigenous to characteristics 3 and 2, we describe the space of cohomology with trivial coefficients. We show that the natural multiplication in this space is very complicated.

KW - Lie superalgebra cohomology

KW - Lie superalgebra deformation

KW - modular Lie superalgebra

UR - http://www.scopus.com/inward/record.url?scp=85161809440&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85161809440&partnerID=8YFLogxK

U2 - 10.3842/SIGMA.2023.031

DO - 10.3842/SIGMA.2023.031

M3 - Article

AN - SCOPUS:85161809440

SN - 1815-0659

VL - 19

JO - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

JF - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

M1 - 031

ER -